Estimating favorability

\[ \begin{equation} F(\mathbf{s}) = \prod_{M=1}^{m}X_m(\mathbf{s}) \end{equation} \]

• Treat \(F(\mathbf{s})\) as binary
• Then \(F(\mathbf{s}) = 1\) if all inputs (\(X_m(\mathbf{s})\)) are suitable
• Then \(F(\mathbf{s}) = 0\) if not

Estimating favorability

\[ \begin{equation} F(\mathbf{s}) = f(w_1X_1(\mathbf{s}), w_2X_2(\mathbf{s}), w_3X_3(\mathbf{s}), ..., w_mX_m(\mathbf{s})) \end{equation} \]

• \(F(\mathbf{s})\) does not have to be binary (could be ordinal or continuous)

• \(X_m(\mathbf{s})\) could also be extended beyond simply ‘suitable/not suitable’

• Adding weights allows incorporation of relative importance

• Other functions for combining inputs (\(X_m(\mathbf{s})\))

Weighted Linear Combinations

\[ \begin{equation} F(\mathbf{s}) = \frac{\sum_{i=1}^{m}w_iX_i(\mathbf{s})}{\sum_{i=1}^{m}w_i} \end{equation} \]

• \(F(s)\) is now an index based of the values of \(X_m(\mathbf{s})\)

• \(w_i\) can incorporate weights of evidence, uncertainty, or different participant preferences

• Dividing by \(\sum_{i=1}^{m}w_i\) normalizes by the sum of weights

Model-driven overlay

\[ \begin{equation} F(\mathbf{s}) = w_0 + \sum_{i=1}^{m}w_iX_i(\mathbf{s}) + \epsilon \end{equation} \]

• If we estimate \(w_i\) using data, we specify \(F(s)\) as the outcome of regression

• When \(F(s)\) is binary → logistic regression

• When \(F(s)\) is continuous → linear (gamma) regression

• When \(F(s)\) is discrete → Poisson regression

• Assumptions about \(\epsilon\) matter!!

Why do we create distribution models?

General analysis situation

From Long

Modeling Presence-Absence Data

Logistic regression

An Example

Inputs from the dismo package

An Example

The sample data

head(pres.abs)

An Example

Building our dataframe

pts.df <- terra::extract(pred.stack, vect(pres.abs), df=TRUE)
head(pts.df)

An Example

Building our dataframe

pts.df[,2:7] <- scale(pts.df[,2:7])
summary(pts.df)

An Example

Looking at correlations

pairs(pts.df[,2:7])

An Example

Looking at correlations

corrplot(cor(pts.df[,2:7]), method = "number")

An Example

Fitting some models

pts.df <- cbind(pts.df, pres.abs$y)
colnames(pts.df)[8] <- "y"
logistic.global <- glm(y~., family=binomial(link="logit"), data=pts.df[,2:8])
logistic.simple <- glm(y ~ MeanAnnTemp + TotalPrecip, family=binomial(link="logit"), data=pts.df[,2:8])
logistic.rich <- glm(y ~ MeanAnnTemp + PrecipWetQuarter + PrecipDryQuarter, family=binomial(link="logit"), data=pts.df[,2:8])

An Example

Checking out the results

summary(logistic.global)

An Example

Checking out the results

summary(logistic.simple)

An Example

Checking out the results

summary(logistic.rich)

An Example

Comparing models

AIC(logistic.global, logistic.simple, logistic.rich)

An Example

Generating predictions

preds <- predict(object=pred.stack, model=logistic.simple)
plot(preds)
plot(pres.pts$geometry, add=TRUE, pch=3, col="blue")
plot(abs.pts$geometry, add=TRUE, pch ="-", col="red")

An Example

Generating predictions

preds <- predict(object=pred.stack, model=logistic.simple, type="response")
plot(preds)
plot(pres.pts$geometry, add=TRUE, pch=3, col="blue")
plot(abs.pts$geometry, add=TRUE, pch ="-", col="red")

An Example

Generating predictions

preds <- predict(object=pred.stack, model=logistic.global, type="response")
plot(preds)
plot(pres.pts$geometry, add=TRUE, pch=3, col="blue")
plot(abs.pts$geometry, add=TRUE, pch ="-", col="red")

An Example

Generating predictions

preds <- predict(object=pred.stack, model=logistic.rich, type="response")
plot(preds)
plot(pres.pts$geometry, add=TRUE, pch=3, col="blue")
plot(abs.pts$geometry, add=TRUE, pch ="-", col="red")

Key assumptions of logistic regression

• Dependent variable must be binary

• Observations must be independent (important for spatial analyses)

• Predictors should not be collinear

• Predictors should be linearly related to the log-odds

• Sample Size

The sampling situation

The Challenge with Background Points

• What constitutes background?

• Not measuring probability, but relative likelihood of occurrence

• Sampling bias affects estimation

• The intercept

\[ \begin{equation} y_{i} \sim \text{Bern}(p_i)\\ \text{link}(p_i) = \mathbf{x_i}'\beta + \alpha \end{equation} \]

Point Process Models

• Poisson Point Process Models model location (not \(y\))

• Number of points expected is given by a rate \(\lambda\)

• Model \(\lambda\) using Poisson regression